We study symmetric tensor decompositions, i.e., decompositions of the form $T = \sum_{i=1}^r u_i^{\otimes 3}$ where $T$ is a symmetric tensor of order 3 and $u_i \in \mathbb{C}^n$.In order to obtain efficient decomposition algorithms, it is necessary to require additional properties from $u_i$. In this paper we assume that the $u_i$ are linearly independent. This implies $r \leq n$,that is, the decomposition of T is undercomplete. We give a randomized algorithm for the following problem in the exact arithmetic model of computation: Let $T$ be an order-3 symmetric tensor that has an undercomplete decomposition.Then given some $T'$ close to $T$, an accuracy parameter $\varepsilon$, and an upper bound B on the condition number of the tensor, output vectors $u'_i$ such that $||u_i - u'_i|| \leq \varepsilon$ (up to permutation and multiplication by cube roots of unity) with high probability. The main novel features of our algorithm are: 1) We provide the first algorithm for this problem that runs in linear time in the size of the input tensor. More specifically, it requires $O(n^3)$ arithmetic operations for all accuracy parameters $\varepsilon =$ 1/poly(n) and B = poly(n). 2) Our algorithm is robust, that is, it can handle inverse-quasi-polynomial noise (in $n$,B,$\frac{1}{\varepsilon}$) in the input tensor. 3) We present a smoothed analysis of the condition number of the tensor decomposition problem. This guarantees that the condition number is low with high probability and further shows that our algorithm runs in linear time, except for some rare badly conditioned inputs. Our main algorithm is a reduction to the complete case ($r=n$) treated in our previous work [Koiran,Saha,CIAC 2023]. For efficiency reasons we cannot use this algorithm as a blackbox. Instead, we show that it can be run on an implicitly represented tensor obtained from the input tensor by a change of basis.

翻译：我们研究对称张量分解，即形如 $T = \sum_{i=1}^r u_i^{\otimes 3}$ 的分解，其中 $T$ 为三阶对称张量且 $u_i \in \mathbb{C}^n$。为获得高效分解算法，需对 $u_i$ 附加额外性质。本文假设 $u_i$ 线性无关，这意味着 $r \leq n$，即 $T$ 的分解为欠完备分解。在精确计算模型下，我们提出如下问题的随机化算法：设 $T$ 为具有欠完备分解的三阶对称张量。给定接近 $T$ 的张量 $T'$、精度参数 $\varepsilon$ 以及张量条件数的上界 $B$，算法输出向量 $u'_i$，使得 $\|u_i - u'_i\| \leq \varepsilon$（在置换和立方根相乘意义下）以高概率成立。本算法的主要创新点包括：1) 首次实现该问题在输入张量规模下的线性时间运行。具体而言，当 $\varepsilon = 1/\text{poly}(n)$ 且 $B = \text{poly}(n)$ 时，仅需 $O(n^3)$ 次算术运算。2) 算法具有鲁棒性，可处理输入张量中逆拟多项式（关于 $n,B,\frac{1}{\varepsilon}$）的噪声。3) 我们给出张量分解问题条件数的平滑分析，保证条件数以高概率保持低值，并进一步证明除少数病态输入外算法均在线性时间内完成。主算法通过归约到我们先前工作 [Koiran,Saha,CIAC 2023] 处理的完备情形 ($r=n$) 实现。出于效率考虑，无法将该算法作为黑箱使用。相反，我们证明可将其运行于通过基变换从输入张量获得的隐式表示张量之上。